>>292200381. There's 126 dungeons. Can't really remember how many were used for the plot, and how many are endgame only.
3. The game's pretty restrictive until you reach the postgame, which shouldn't take too long. The postgame's when it starts to open up, but there's no real plot to it. You mostly just recruit members through the connection orb system and fight legendary bosses.
4. Postgame only, you talk to Fennekin and Braixen to recruit them, and Delphox is recruited by connecting with Skitty.